Thermodynamics, Heat Transfer, And Fluid Flow. V.1. Thermodynamics, страница 16

This is given by:qr = To ∆s in units of Btu/lbm,(1-37)where To is the average heat sink temperature of 520°R. The available energy (A.E.) for theCarnot cycle may be given as:A.E. = qs - qr.(1-38)Substituting equation 1-37 for qr gives:A.E. = qs - To ∆sin units of Btu/lbm.(1-39)and is equal to the area of the shadedregion labeled available energy inFigure 28 between the temperatures1962° and 520°R. From Figure 28 itcan been seen that any cycle operatingat a temperature of less than 1962°Rwill be less efficient. Note that bydeveloping materials capable ofwithstanding the stresses above1962°R, we could greatly add to theenergy available for use by the plantcycle.From equation 1-37, one can see whythe change in entropy can be definedas a measure of the energy unavailableto do work.

If the temperature of theheat sink is known, then the change inentropy does correspond to a measureof the heat rejected by the engine.Rev. 0Figure 28Page 85Carnot CycleHT-01SECOND LAW OF THERMODYNAMICSThermodynamicsFigure 29 Carnot Cycle vs. Typical Power Cycle Available EnergyHT-01Page 86Rev. 0ThermodynamicsSECOND LAW OF THERMODYNAMICSFigure 29 is a typical power cycle employed by a fossil fuel plant. The working fluid is water,which places certain restrictions on the cycle. If we wish to limit ourselves to operation at orbelow 2000 psia, it is readily apparent that constant heat addition at our maximum temperatureof 1962°R is not possible (Figure 29, 2’ to 4).

In reality, the nature of water and certainelements of the process controls require us to add heat in a constant pressure process instead(Figure 29, 1-2-3-4). Because of this, the average temperature at which we are adding heat isfar below the maximum allowable material temperature.As can be seen, the actual available energy (area under the 1-2-3-4 curve, Figure 29) is less thanhalf of what is available from the ideal Carnot cycle (area under 1-2’-4 curve, Figure 29)operating between the same two temperatures. Typical thermal efficiencies for fossil plants areon the order of 40% while nuclear plants have efficiencies of the order of 31%. Note that thesenumbers are less than 1/2 of the maximum thermal efficiency of the ideal Carnot cycle calculatedearlier.Figure 30 shows a proposed Carnot steam cycle superimposed on a T-s diagram.

As shown, ithas several problems which make it undesirable as a practical power cycle. First a great deal ofpump work is required to compress a two phase mixture of water and steam from point 1 to thesaturated liquid state at point 2. Second, this same isentropic compression will probably resultin some pump cavitation in the feed system. Finally, a condenser designed to produce a twophase mixture at the outlet (point 1) would pose technical problems.Figure 30 Ideal Carnot CycleRev. 0Page 87HT-01SECOND LAW OF THERMODYNAMICSThermodynamicsEarly thermodynamic developments were centered around improving the performance ofcontemporary steam engines.

It was desirable to construct a cycle that was as close to beingreversible as possible and would better lend itself to the characteristics of steam and processcontrol than the Carnot cycle did. Towards this end, the Rankine cycle was developed.The main feature of the Rankine cycle, shown in Figure 31, is that it confines the isentropiccompression process to the liquid phase only (Figure 31 points 1 to 2). This minimizes theamount of work required to attain operating pressures and avoids the mechanical problemsassociated with pumping a two-phase mixture. The compression process shown in figure 31between points 1 and 2 is greatly exaggerated*.

In reality, a temperature rise of only 1°F occursin compressing water from 14.7 psig at a saturation temperature of 212°F to 1000 psig.Figure 31 Rankine Cycle* The constant pressure lines converge rapidly in the subcooled or compressedliquid region and it is difficult to distinguish them from the saturated liquid linewithout artificially expanding them away from it.In a Rankine cycle available and unavailable energy on a T-s diagram, like a T-s diagram of aCarnot cycle, is represented by the areas under the curves. The larger the unavailable energy,the less efficient the cycle.HT-01Page 88Rev. 0ThermodynamicsSECOND LAW OF THERMODYNAMICSFigure 32Rankine Cycle With Real v.s.

IdealFrom the T-s diagram (Figure 32) it can also be seen that if an ideal component, in this case theturbine, is replaced with a non-ideal component, the efficiency of the cycle will be reduced. Thisis due to the fact that the non-ideal turbine incurs an increase in entropy which increases the areaunder the T-s curve for the cycle. But the increase in the area of available energy (3-2-3’,Figure 32) is less than the increase in area for unavailable energy (a-3-3’-b, Figure 32).Figure 33 Rankine Cycle Efficiencies T-sRev. 0Page 89HT-01SECOND LAW OF THERMODYNAMICSThermodynamicsThe same loss of cycle efficiency can be seen when two Rankine cycles are compared (see Figure33).

Using this type of comparison, the amount of rejected energy to available energy of onecycle can be compared to another cycle to determine which cycle is the most efficient, i.e. hasthe least amount of unavailable energy.An h-s diagram can also beused to compare systems andhelp determine theirefficiencies.Like the T-sdiagram, the h-s diagram willshow (Figure 34) thatsubstituting non-idealcomponents in place of idealcomponents in a cycle, willresult in the reduction in thecycles efficiency.This isbecause a change in enthalpy(h) always occurs when workis done or heat is added orremoved in an actual cycle(non-ideal).This deviationFigure 34 h-s Diagramfrom an ideal constant enthalpy(vertical line on the diagram)allows the inefficiencies of the cycle to be easily seen on a h-s diagram.Typical Steam CycleFigure 35 shows a simplified version of the major components of a typical steam plant cycle.This is a simplified version and does not contain the exact detail that may be found at mostpower plants.

However, for the purpose of understanding the basic operation of a power cycle,further detail is not necessary.The following are the processes that comprise the cycle:HT-011-2:Saturated steam from the steam generator is expanded in the high pressure (HP)turbine to provide shaft work output at a constant entropy.2-3:The moist steam from the exit of the HP turbine is dried and superheated in themoisture separator reheater (MSR).3-4:Superheated steam from the MSR is expanded in the low pressure (LP) turbine toprovide shaft work output at a constant entropy.Page 90Rev.

0ThermodynamicsSECOND LAW OF THERMODYNAMICS4-5:Steam exhaust from the turbine is condensed in the condenser in which heat istransferred to the cooling water under a constant vacuum condition.5-6:The feedwater is compressed as a liquid by the condensate and feedwater pumpand the feedwater is preheated by the feedwater heaters.6-1:Heat is added to the working fluid in the steam generator under a constantpressure condition.Figure 35Typical Steam CycleThe previous cycle can also be represented on a T-s diagram as was done with the ideal Carnotand Rankine cycles.

This is shown in Figure 36. The numbered points on the cycle correspondto the numbered points on Figure 36.It must be pointed out that the cycle we have just shown is an ideal cycle and does not exactlyrepresent the actual processes in the plant. The turbine and pumps in an ideal cycle are idealpumps and turbines and therefore do not exhibit an increase in entropy across them. Real pumpsand turbines would exhibit an entropy increase across them.Rev. 0Page 91HT-01SECOND LAW OF THERMODYNAMICSFigure 36ThermodynamicsSteam Cycle (Ideal)Figure 37 is a T-s diagram of a cycle which more closely approximates actual plant processes.The pumps and turbines in this cycle more closely approximate real pumps and turbines and thusexhibit an entropy increase across them.

Additionally, in this cycle, a small degree of subcoolingis evident in the condenser as shown by the small dip down to point 5. This small amount ofsubcooling will decrease cycle efficiency since additional heat has been removed from the cycleto the cooling water as heat rejected.